Patients are allocated to indenpendent treatments A sub 1, A sub 2, and we observe the subjects survival times which are assumed to have the probability density function f (x; mu sub i, sigma sub i) under the treatment A sub i with f (x; mu, sigma) = sigma minus 1 exp (-(x-mu)/sigma) for x greater than mu, i = 1,2. We let mu sub i epsilon equal from minus infinity to infinity, sigma sub i epsilon equal zero to infinity, and i equaling 1,2. The problems of constructing fixed width confidence intervals of mu sub 1, a minimum risk permit estimator of theta sub 1, a minimum risk estimator of sigma sub 1 for the treatment A sub 1 are discussed. For these series of one-sample problems for A sub 1 (similarly A sub 2), the modified two-stage, sequential and three-stage procedures are utilized. The problem of constructing fixed-width confidence intervals for mu sub 1 - mu sub 2, a minimum risk point estimator of theta sub 1- theta sub 2 are also studied. In the two sample case when the form of the density function is unspecified, a fixed width confidence interval is proposed for mu sub 1- mu sub 2. Again, modified two-stage, sequential and thru-stage procedures are utilized. Extensive uses of the computer simulations will be made to study the moderate sample performances of all the proposed procedures. Various theoretical expansions of some characteristics of our procedures will be studied via non-linear renewal theory. This comprehensive study will fill many significant gaps in the existing literature of sequential exponential clinical trials, and also it will definitely open up many other avenues of interesting research in the future.